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We study the classic $k$-median and $k$-means clustering objectives in the beyond-worst-case scenario. We consider three well-studied notions of structured data that aim at characterizing real-world inputs: Distribution Stability…
Partial Maximum Satisfiability (PMS) and Weighted Partial Maximum Satisfiability (WPMS) generalize Maximum Satisfiability (MaxSAT), with broad real-world applications. Recent advances in Stochastic Local Search (SLS) algorithms for solving…
This work presents two new algorithms for performing constraint satisfaction. The first algorithm presented, DMaxWalkSat, is a constraint solver specialized for solving dynamic, weighted constraint satisfaction problems. The second…
The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its…
We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the…
Local Search is one of the fundamental approaches to combinatorial optimization and it is used throughout AI. Several local search algorithms are based on searching the k-exchange neighborhood. This is the set of solutions that can be…
We introduce a highly structured family of hard satisfiable 3-SAT formulas corresponding to an ordered spin-glass model from statistical physics. This model has provably "glassy" behavior; that is, it has many local optima with large energy…
We study the behavior of ASAT, a heuristic for solving satisfiability problems by stochastic local search near the SAT/UNSAT transition. The heuristic is focused, i.e. only variables in unsatisfied clauses are updated in each step, and is…
Depth-3 circuit lower bounds and $k$-SAT algorithms are intimately related; the state-of-the-art $\Sigma^k_3$-circuit lower bound and the $k$-SAT algorithm are based on the same combinatorial theorem. In this paper we define a problem which…
This paper proposes a new framework for providing approximation guarantees of local search algorithms. Local search is a basic algorithm design technique and is widely used for various combinatorial optimization problems. To analyze local…
Random $K$-satisfiability ($K$-SAT) is a paradigmatic model system for studying phase transitions in constraint satisfaction problems and for developing empirical algorithms. The statistical properties of the random $K$-SAT solution space…
Let F be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-constructive arguments show that F is satisfiable for clause/variable ratios m/n< r(k)~2^k ln 2 with high probability. Yet no efficient algorithm is…
Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the…
Local search algorithms applied to optimization problems often suffer from getting trapped in a local optimum. The common solution for this deficiency is to restart the algorithm when no progress is observed. Alternatively, one can start…
A previously developed quantum search algorithm for solving 1-SAT problems in a single step is generalized to apply to a range of highly constrained k-SAT problems. We identify a bound on the number of clauses in satisfiability problems for…
Quantum computer algorithms can exploit the structure of random satisfiability problems. This paper extends a previous empirical evaluation of such an algorithm and gives an approximate asymptotic analysis accounting for both the average…
Local search has been demonstrated as an efficient approach for two practical generalizations of the MaxSAT problem, namely Partial MaxSAT (PMS) and Weighted PMS (WPMS). In this work, we observe that most local search (W)PMS solvers usually…
Dominating Set is a well-known combinatorial optimization problem which finds application in computational biology or mobile communication. Because of its $\mathrm{NP}$-hardness, one often turns to heuristics for good solutions. Many such…
In the last decades, many efforts have focused on analyzing typical-case hardness in optimization and inference problems. Some recent work has pointed out that polynomial algorithms exist, running with a time that grows more than linearly…
We propose a unifying framework for smoothed analysis of combinatorial local optimization problems, and show how a diverse selection of problems within the complexity class PLS can be cast within this model. This abstraction allows us to…